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Find the reciprocal of each number. Then check that the product of each number and its reciprocal is 1 .

  1. 4 9
  2. 1 6
  3. 14 5
  4. 7

Solution

To find the reciprocals, we keep the sign and invert the fractions.

Find the reciprocal of 4 9 . The reciprocal of 4 9 is 9 4 .
Check:
Multiply the number and its reciprocal. 4 9 9 4
Multiply numerators and denominators. 36 36
Simplify. 1
Find the reciprocal of - 1 6 . - 6 1
Simplify. - 6
Check: - 1 6 ( - 6 )
1
Find the reciprocal of - 14 5 . - 5 14
Check: - 14 5 ( - 5 14 )
70 70
1
Find the reciprocal of 7 .
Write 7 as a fraction. 7 1
Write the reciprocal of 7 1 . 1 7
Check: 7 ( 1 7 )
1
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Find the reciprocal:

  1. 5 7
  2. 1 8
  3. 11 4
  4. 14

  1. 7 5
  2. −8
  3. 4 11
  4. 1 14

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Find the reciprocal:

  1. 3 7
  2. 1 12
  3. 14 9
  4. 21

  1. 7 3
  2. 12
  3. 9 14
  4. 1 21

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In a previous chapter, we worked with opposites and absolute values. [link] compares opposites, absolute values, and reciprocals.

Opposite Absolute Value Reciprocal
has opposite sign is never negative has same sign, fraction inverts

Fill in the chart for each fraction in the left column:

Number Opposite Absolute Value Reciprocal
3 8
1 2
9 5
−5

Solution

To find the opposite, change the sign. To find the absolute value, leave the positive numbers the same, but take the opposite of the negative numbers. To find the reciprocal, keep the sign the same and invert the fraction.

Number Opposite Absolute Value Reciprocal
3 8 3 8 3 8 8 3
1 2 1 2 1 2 2
9 5 9 5 9 5 5 9
−5 5 5 1 5
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Fill in the chart for each number given:

Number Opposite Absolute Value Reciprocal
5 8
1 4
8 3
−8


No alt text

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Fill in the chart for each number given:

Number Opposite Absolute Value Reciprocal
4 7
1 8
9 4
−1
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Divide fractions

Why is 12 ÷ 3 = 4 ? We previously modeled this with counters. How many groups of 3 counters can be made from a group of 12 counters?

Four red ovals are shown. Inside each oval are three grey circles.

There are 4 groups of 3 counters. In other words, there are four 3 s in 12 . So, 12 ÷ 3 = 4 .

What about dividing fractions? Suppose we want to find the quotient: 1 2 ÷ 1 6 . We need to figure out how many 1 6 s there are in 1 2 . We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown in [link] . Notice, there are three 1 6 tiles in 1 2 , so 1 2 ÷ 1 6 = 3 .

A rectangle is shown, labeled as one half. Below it is an identical rectangle split into three equal pieces, each labeled as one sixth.
Doing the Manipulative Mathematics activity "Model Fraction Division" will help you develop a better understanding of dividing fractions.

Model: 1 4 ÷ 1 8 .

Solution

We want to determine how many 1 8 s are in 1 4 . Start with one 1 4 tile. Line up 1 8 tiles underneath the 1 4 tile.

A rectangle is shown, labeled one fourth. Below it is an identical rectangle split into two equal pieces, each labeled as one eighth.

There are two 1 8 s in 1 4 .

So, 1 4 ÷ 1 8 = 2 .

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Model: 2 ÷ 1 4 .

Solution

We are trying to determine how many 1 4 s there are in 2 . We can model this as shown.

Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into four pieces. Each of the eight pieces is labeled as one fourth.

Because there are eight 1 4 s in 2 , 2 ÷ 1 4 = 8 .

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Let’s use money to model 2 ÷ 1 4 in another way. We often read 1 4 as a ‘quarter’, and we know that a quarter is one-fourth of a dollar as shown in [link] . So we can think of 2 ÷ 1 4 as, “How many quarters are there in two dollars?” One dollar is 4 quarters, so 2 dollars would be 8 quarters. So again, 2 ÷ 1 4 = 8 .

A picture of a United States quarter is shown.
The U.S. coin called a quarter is worth one-fourth of a dollar.

Using fraction tiles, we showed that 1 2 ÷ 1 6 = 3 . Notice that 1 2 · 6 1 = 3 also. How are 1 6 and 6 1 related? They are reciprocals. This leads us to the procedure for fraction division.

Fraction division

If a , b , c , and d are numbers where b 0 , c 0 , and d 0 , then

a b ÷ c d = a b · d c

To divide fractions, multiply the first fraction by the reciprocal of the second.

Practice Key Terms 2

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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